Since we want to simulate the interaction of the magnetospheres of
two neutron stars in a close contact binary, we also need to model the stars as well as
their orbital and rotational motion.
For simplicity, we adopt spherical conductors with a radius of \(14.76\,\rm km\)  on a circular orbit. Our results are insensitive to small variations of the radius
within currently favored uncertainties, see e.g. [1]}.
In this quasi-adiabatic inspiral, we neglect
changes in the orbital separation, \(a\) , which is a good approximation at moderate
distances, since the separation changes only slowly compared to the orbital
period [2]}.
Following Refs. [3]} and [4]}, we model the
rotation of the sphere in terms of a local spin frequency \(\omega _{1/2}\)  of
the spheres around their axis, and a global orbital motion in terms of the
orbital frequency \(\Omega \) . We can then split the advected fluid velocity
\(u^i/u^0 = v^i - \beta ^i\)  into a global shift and three-dimensional
velocity part \(v^i\) . For these, we adopt
\(\beta ^i =&\, - \varepsilon ^{ijk} \Omega \hat{z}_j x_k, \\v^i =&\, \varepsilon ^{ijk} \omega \hat{z}_j \left( x_k - x^{ 1/2}_k\right) - \varepsilon ^{ijk} \Omega \hat{z}_jx^{1/2}_k, \\u^i/u^0 =&\, v^i - \beta ^i \nonumber \\=&\, \varepsilon ^{ijk} \left( \Omega + \omega \right) \hat{z}_j\left( x_k - x^{ 1/2}_k \right) ,\) 
